Show that the equation to the parallel line mid-way between the parallel lines
`ax+by+c_(1)=0andax+by+c_(2)=0` is `ax+by+(c_(1)+c_(2))/2=0`

Find the distance between the parallel lines ax+by+c=0 and ax+by+d=0

Prove that distance between two parallel lines ax+by+c_1=0 and ax+by+c_2=0 is given by |c_1-c_2|/(sqrt(a^2+b^2))

Show that the distance between the parallel lines ax+by+c=0 and k(ax+by)+d=0 is |(c-(d)/(k))/(sqrt(a^(2)+b^(2)))|

Find the equation of the line through (-2, -1) and parallel to line x=0 .

Show that the equation to the plane containing the line y/b+ z/c =1, x=0, and parallel to the line x/a -z/c=1, y=0is x/a-y/b -z/c+1=0, and if 2d is the shortest, distance between given lines, prove that (1)/(d ^(2)) = (1)/(a ^(2)) + (1)/(b ^(2)) + (1)/(c ^(2)).

Consider the following statements : 1. The distance between the lines y=mx+c_(1) and y=mx+c_(2) is (|c_(1)-c_(2)|)/sqrt(1-m^(2)) . 2. The distance between the lines ax+by+c_(1) and ax+by+c_(2)=0 is (|c_(1)-c_(2)|)/sqrt(a^(2)+b^(2)) . 3. The distance between the lines x=c_(1) and x=c_(2) is |c_(1)-c_(2)| . Which of the above statements are correct ?

If a, c, b are in G.P then the line ax + by + c= 0

If the origin lies in the acute angle between the lines a_1 x + b_1 y + c_1 = 0 and a_2 x + b_2 y + c_2 = 0 , then show that (a_1 a_2 + b_1 b_2) c_1 c_2 lt0 .

Find the relation between a and b if the lines 3x-by+5=0 and ax+y=2 parallel.

The equation of the plane through the line of intersection of the planes ax + by+cz + d= 0 and a'x + b'y+c'z + d'= 0 parallel to the line y=0 and z=0 is